Interpretation of aeromagnetic data using pseudo gravity gradient
4
EGM 2010 International Workshop Adding new value to Electromagnetic, Gravity and Magnetic Methods for Exploration Capri, Italy, April 11-14, 2010 Interpretation of aeromagnetic data using pseudo gravity gradient tensor decomposition M. Beiki, L. B. Pedersen Department of Earth Sciences, Uppsala University, Uppsala, Sweden Summary The eigenvectors of the symmetric gravity gradient tensor can be used to estimate the position of the source body as well as its strike direction. For a given measurement point, the eigenvector corresponding to the maximum eigenvalue points approximately towards the center of mass of the causative body. For a collection of measurement points a robust least squares procedure is used to estimate the source point as that point which has the smallest sum of square distances to the lines defined by the eigenvectors and the measurement positions. The strike direction of the source can be estimated from the direction of the eigenvectors corresponding to the smallest eigenvalue for quasi 2D structures. Assuming that the magnetization direction is known, the pseudo gravity gradient tensor derived from magnetic field data has the same properties. Introduction The Gravity Gradient Tensor (GGT) is a multi-component gravity system computed from the second derivatives of the earth’s gravitational potential in the three Cartesian directions (Zhdanov et al., 2004). GGT data can be measured either using airborne or marine platforms. In recent years, the applications of GGT data in hydrocarbon exploration, mineral exploration and structural geology have increased considerably (Fedi, et al., 2005). Pedersen and Rasmussen (1990) studied gradient tensors of gravity and magnetic fields and introduced scalar invariants to indicate their dimensionality. They also showed that the maximum eigenvalue and corresponding eigenvector of the GGT is related to the center of mass for a simple point source. In this abstract, we present a new method to locate the causative bodies from a collection of eigenvectors of the GGT using robust least squares. The method is tested on the Pseudo Gravity Gradient Tensor (PGGT) calculated from total magnetic field data. The method is applied to aeromagnetic data from the well-known Siljan impact structure in Sweden. Methodology The gravity gradient tensor, can be written as: Γ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = zz yz xz yz yy xy xz xy xx g g g g g g g g g z U y z U x z U z y U y U x y U z x U y x U x U 2 2 2 2 2 2 2 2 2 2 2 2 Γ (2) Outside of the sources the gravitational potential, U satisfies Laplace’s equation , and hence the trace of the tensor, () 0 2 = ∇ r U ( ) 0 33 22 11 Γ + Γ + = Γ = Γ Trace and since is symmetric it only contains Γ
Transcript of Interpretation of aeromagnetic data using pseudo gravity gradient
EGM 2010 International WorkshopAdding new value to Electromagnetic, Gravity and Magnetic Methods for Exploration
Capri, Italy, April 11-14, 2010
Interpretation of aeromagnetic data using pseudo gravity gradient tensor decomposition
M. Beiki, L. B. Pedersen Department of Earth Sciences, Uppsala University, Uppsala, Sweden
Summary The eigenvectors of the symmetric gravity gradient tensor can be used to estimate the position of the source body as well as its strike direction. For a given measurement point, the eigenvector corresponding to the maximum eigenvalue points approximately towards the center of mass of the causative body. For a collection of measurement points a robust least squares procedure is used to estimate the source point as that point which has the smallest sum of square distances to the lines defined by the eigenvectors and the measurement positions. The strike direction of the source can be estimated from the direction of the eigenvectors corresponding to the smallest eigenvalue for quasi 2D structures. Assuming that the magnetization direction is known, the pseudo gravity gradient tensor derived from magnetic field data has the same properties.
Introduction The Gravity Gradient Tensor (GGT) is a multi-component gravity system computed from the second derivatives of the earth’s gravitational potential in the three Cartesian directions (Zhdanov et al., 2004). GGT data can be measured either using airborne or marine platforms. In recent years, the applications of GGT data in hydrocarbon exploration, mineral exploration and structural geology have increased considerably (Fedi, et al., 2005). Pedersen and Rasmussen (1990) studied gradient tensors of gravity and magnetic fields and introduced scalar invariants to indicate their dimensionality. They also showed that the maximum eigenvalue and corresponding eigenvector of the GGT is related to the center of mass for a simple point source. In this abstract, we present a new method to locate the causative bodies from a collection of eigenvectors of the GGT using robust least squares. The method is tested on the Pseudo Gravity Gradient Tensor (PGGT) calculated from total magnetic field data. The method is applied to aeromagnetic data from the well-known Siljan impact structure in Sweden.
Methodology The gravity gradient tensor, can be written as: Γ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂∂
∂∂∂
∂∂∂
∂∂
∂∂∂
∂∂∂
∂∂∂
∂∂
=
zzyzxz
yzyyxy
xzxyxx
ggg
ggg
ggg
zU
yzU
xzU
zyU
yU
xyU
zxU
yxU
xU
2
2
2
222
222
222
Γ
(2)
Outside of the sources the gravitational potential, U satisfies Laplace’s equation , and hence the trace of the tensor,
( ) 02 =∇ rU( ) 0332211Γ +Γ + =Γ=ΓTrace and since is symmetric it only contains Γ
EGM 2010 International WorkshopAdding new value to Electromagnetic, Gravity and Magnetic Methods for Exploration
Capri, Italy, April 11-14, 2010
five independent components. ( )zyx ggg ,,=g is the gravity vector and superscript T means transposition. , being a symmetric matrix can be diagonalized as: Γ
ΛΓVVT = (3)
where and contains eigenvectors and eigenvalues, respectively.
Physically, equation 3 means that at any observation point, one can find a new coordinate system with axes along the eigenvectors in which the gradient tensor is on diagonal form (Pedersen and Rasmussen, 1990; Mikhailov et al., 2007). Pedersen and Rasmussen (1990) introduced the following three invariants:
[ ]321 vvvV =⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
3
2
1
Λλ
λλ
000000
( ) 03210 =++== λλλΓTraceI (4)
3132212
132
232
123311332222111 λλλλλλ ++=Γ−Γ−Γ−ΓΓ+ΓΓ+ΓΓ=I (5) and
( ) 3212 det λλλ== ΓI (6)
They showed that the invariant ratio, I defined as 31
22
427
III −= lies between zero and unity for any
potential field. If the causative body is strictly 2D then I is equal to zero for all measurement points. When the causative body as seen from the observation point looks more and more 3D like, then I increases and eventually approaches unity. Following Pedersen and Rasmussen (1990) 1λ is the maximum eigenvalue of the tensor and
( Rzzyyxx /,, 0001 )−−−=v is the corresponding eigenvector, where ( )zyxP ,, and represent the known observation point and the unknown ‘’center of the mass’’, respectively and
( )0000 ,, zyxP
( ) ( ) ( )202
02
0 zzyyxxR −+−+−= is distance between and . P 0PThe equation denotes the straight line passing through the observation point
in the direction of . The distance from to this line is given by ( ) 01 =×′− vrr
( zyxP ,, ) 1v 0P( )
1
01
vrrv −×
=d (7)
where 11 =v (Polyanin and Manzhirov, 2007). For two or more observation points the least squares solution minimizing the squared sum of distances to the lines will be taken as the location of the equivalent source point. Then, the distance from the source point to the line passing through each observation point parallel to the eigenvector at can be determined by minimizing the expression
( iiii zyxP ,, ) 1v iP
∑=
=N
iidQ
1
2 (8)
where is the number of observation points in the window. NAssuming that maxima or minima of the first vertical derivative of the vertical component of gravity, , occurs approximately above the center of mass for positive or negative anomalies, respectively, a window centered on the located maximum or minimum is formed.
zzg
In our algorithm, we increase the window size until the standard error reaches a minimum or the window size exceeds a predefined limit, and the estimated source location is then taken as the final source location.
EGM 2010 International WorkshopAdding new value to Electromagnetic, Gravity and Magnetic Methods for Exploration
Capri, Italy, April 11-14, 2010
For a simple line source the eigenvectors corresponding to the minimum and intermediate eigenvalues point in the strike direction and the direction of the horizontal gradient of the field, respectively. For each single measurement point the strike direction can be determined from the direction of the eigenvector corresponding to the smallest eigenvalue. Baranov (1957) introduced the pseudogravity transformation based on Poisson’s relation. This transformation relates the total field magnetic anomaly to the vertical component of the gravity field, . Subsequently, the horizontal components, and as well as their horizontal and vertical derivatives, can be calculated to form the complete PGGT. Assuming that there is no remanent magnetization, the PGGT calculated from measured magnetic field shows the same properties.
zg xg yg
Aeromagnetic data In the 1980’s, the Siljan impact structure was covered with aeromagnetic measurements conducted by Geological Survey of Sweden (SGU). The nominal height of the aircraft was 30 m above the ground with a sampling interval of almost 40 m along the flight lines. The flight lines were flown in E-W direction with spacing of 200 m. To avoid aliasing effects, a low pass filter with cut off wave length of 400 m was applied to data and then the data was re-sampled with a new sampling interval of 200 m along the flight lines. The new grid contains the same Nyquist frequency along both directions. Figure 1 shows the gridded total magnetic anomaly with cell size of 50 m after applying the low pass filter and re-sampling the filtered data.
EGM 2010 International WorkshopAdding new value to Electromagnetic, Gravity and Magnetic Methods for Exploration
Capri, Italy, April 11-14, 2010
5.0<IFigure 1. a) Detected strike directions of anomalies with dimensionality, , b) estimated location
of causative bodies based on their depths to the center of mass.
IThe PGGT components are calculated as described earlier to indicate the dimensionality and locate the geological features. To avoid high frequency anomalies which cause undesired local maxima of , the magnetic field data is upward continued to 50 m prior any estimation of the depths and strikes. The estimated strike directions of quasi 2D bodies (
zzg5.0<I ) are shown in Figure 1a.
The estimated depths of causative bodies using the presented algorithm are shown in Figure 1b after upward continuation by 50 m. The solutions are shown based on their estimated depths regardless of the calculated dimensionality.
Conclusions We have described a new method to locate causative bodies using eigenvectors acquired from the GGT or PGGT tensor. We use a robust least squares technique to estimate the intersection point of eigenvectors. It’s also shown that for a pseudo 2D body, the third eigenvectors are in direction of the strike of the field when second eigenvectors are directed toward the maximum gradient of the field. The estimated location of the source for 3D causative bodies is center of mass while for 2D bodies the estimated depth is between center and the top surface of the body. The detected strikes for 2D geological features in the Siljan impact area, is incredibly comparable to geological information. As we showed earlier, for a single 3D body, our method provides a unique solution located at the center of the body. As we use maxima or minima of , solutions are only estimated above the causative bodies and they may not be estimated in areas which are free of causative bodies. Implementation of the introduced method is very quick and can be carried out by users easily.
zzg
Acknowledgements Authors thank Geological Survey of Sweden (SGU) for permission to use magnetic data from Siljan impact structure in this research. We would also like to thank Dr. Mehrdad Bastani for his useful comments.
References Baranov, V., 1957, A new method for interpretation of aeromagnetic maps: Pseudo-gravimetric anomalies: Geophysics, 22, 359-383. Fedi, M., L. Ferrantini, G. Florio, I. Giori, and F. Italiano, 2005, Understanding the structural setting in the southern Apennines (Italy): insight from Gravity Gradient Tensor: Tectonophysics, 397, no.1, 21-36. Mikhailov, V., G. Pajot, M. Diament, and A. Price, 2007, Tensor deconvolution: A method to locate equivalent sources from full tensor gravity data: Geophysics, 72, no.5, 161-169. Pedersen, L. B., and T. M. Rasmussen, 1990, The gradient tensor of potential anomalies: Some implications on data collection and data processing of maps: Geophysics, 55, 1558-1566. Polyanin, A. D., and A. V. Manzhirov, 2007, Handbook of mathematics for engineers and scientists: Chapman & Hall/CRC. Zhdanov, M. S., R. Ellis, and S. Mukherjee, 2004, Three-dimensional regularized focusing inversion of gravity gradient tensor component data: Geophysics, 69, 925-937.
